Is a Delta-Neutral Position Also Gamma-Neutral?

Not necessarily. A Delta-neutral position simply has a net Delta of zero, but it can have a positive or negative net Gamma.

For instance, a short straddle is Delta-neutral but has negative Gamma. A long straddle is also Delta-neutral but has positive Gamma.

Gamma-neutrality is a separate condition that requires the net Gamma of the portfolio to also be zero, which is a higher level of hedging.